Stabilization of a transmission wave plate equation

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Stabilization of a transmission wave/plate equation Kaıs Ammari ? and Serge Nicaise † Abstract. We consider a stabilization problem, for a model arising in the control of noise, coupling the damped wave equation with a damped Kirchoff plate equation. We prove an expo- nential stability result under some geometric condition. Our method is based on an identity with multipliers that allows to show an appropriate energy estimate. Keywords: Wave/plate equation, transmission, boundary stabilization, multiplier. AMS 2000 subject classification: 35B37, 35B40, 93B07, 93D15. 1 Introduction and main results In this paper we consider the stabilization of a system coupling the wave equation with a Kirchhoff system (see [3] for the unidimensional model) damped through a dissipation law on the Kirchhoff system and on the wave system. More precisely we consider a bounded domain ? of IR2 with a Lipschitz boundary such that ?¯ = ?¯1 ? ?¯2, where ?i, i = 1, 2 are bounded domains with a Lipschitz boundary such that ?1 ? ?2 = ?. We then denote by I the interior of ?¯1 ? ?¯2, that is called the interface between ?1 and ?2. For i = 1 or 2, we also set ?i = ∂?i \ I¯, the “exterior” boundary of ?i. We consider the wave equation in ?1 coupled with the Kirchhoff system in ?2, more precisely we consider the following system: ∂2t u1(x, t)?∆u1(x, t) = 0, in ?1 ? (0,+

?u1 ?

lumer-phillips theorem

then there

?v1 ?∆u1

†universite de valenciennes et du hainaut cambresis

exists ?

plate equation

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Stabilization

a transmission
equation

∗
KaısAmmariand

wave/plate

†
Serge Nicaise

Abstract.We consider a stabilization problem, for a model arising in the control of noise,
coupling the damped wave equation with a damped Kirchoﬀ plate equation.We prove an
exponential stability result under some geometric condition.Our method is based on an identity with
multipliers that allows to show an appropriate energy estimate.

Keywords:Wave/plate equation, transmission, boundary stabilization, multiplier.
AMS 2000 subject classiﬁcation:35B37, 35B40, 93B07, 93D15.

Introductionandmainresults

In this paper we consider the stabilization of a system coupling the wave equation with a
Kirchhoﬀ system (see [3] for the unidimensional model) damped through a dissipation law
on the Kirchhoﬀ system and on the wave system.More precisely we consider a bounded
2
¯ ¯¯
domain Ω of IRwith a Lipschitz boundary such that Ω = Ω1∪Ω2, where Ωi, i= 1,2 are
bounded domains with a Lipschitz boundary such that

Ω1∩Ω2=∅.
¯ ¯
We then denote byIthe interior of Ω1∩Ω2, that is called the interface between Ω1and Ω2.
¯
Fori= 1 or 2, we also set Γi=∂Ωi\I, the “exterior” boundary of Ωi.
We consider the wave equation in Ω1coupled with the Kirchhoﬀ system in Ω2, more
precisely we consider the following system:

2
x, t) = 0,in Ω1×(0,+∞),
∂tu1(x, t)−Δu1(
2 2
∂ ut) + Δu(x, t) = 0,in Ω×(0,+∞),
t2(x,2 2
0 1
(x), ∂ u(x,0) =u(x),in Ω, i= 1,2,
ui(x,0) =ui ti i i
u1=u2,B1u2= 0,B2u2=∂ν1u1onI×(0,+∞),
∂ν1u1=−α1u1−∂tu1on Γ1×(0,+∞),
B1u2=−β∂ν2u2−∂ν2∂tu2on Γ2×(0,+∞),
B2u2=α2u2+∂tu2on Γ2×(0,+∞),

(1.1)
(1.2)
(1.3)
(1.4)
(1.5)
(1.6)
(1.7)

whereνi= (νi1, νi2) is the unit normal vector of∂Ωipointing towards the exterior of
Ωi, i= 1,2,andτi= (−νi2, νi1) is the unit tangent vector along∂Ωi. Wefurther denote by
1
∂(resp.∂,∂) the normal (resp.tangent, time) derivative.The constantµ∈(0,) is
νiτit
2
the Poisson coeﬃcient and the boundary operatorBj, j= 1,2 are deﬁned on∂Ω2as follows:
22 2
∂ y∂ y∂ y
2 2
−ν
B1y= Δy+ (1−µ) 2ν21ν22−ν21 22,
2 2
∂x1∂x2∂x ∂x
2 1
∗
D´partement de Math´matiques, Facult´ des Sciences de Monastir, 5019 Monastir, Tunisie,email:
kais.ammari@fsm.rnu.tn
†
Universit´ de Valenciennes et du Hainaut Cambr´sis, LAMAV, FR CNRS 2956, Le Mont Houy, 59313
Valenciennes Cedex 9, France,email : snicaise@univ-valenciennes.fr